One paradigm used in understanding the control of morphogenetic events is the concept of positional information, where sub-organismic components (such as cells) act in response to positional cues. It is important to determine what kinds of spatiotemporal patterns may be obtained by such a method, and what the characteristics of such a morphogenetic process might be. This paper presents a computer model of morphogenesis based on gene activity driven by interpreting a positional information field. In this model, the interactions of mutually regulating developmental genes are viewed as a map from R2 to R2, and are modeled by the complex number algebra. Functions in complex variables are used to simulate genetic interactions resulting in position-dependent differentiation. This is shown to be equivalent to computing modified Julia sets, and is seen to be sufficient to produce a very rich set of morphologies which are similar in appearance and several important characteristics to those of real organisms. The properties of this model can be used to study the potential role of fields and positional information as guiding factors in morphogenesis, as the model facilitates the study of static images, time-series (movies) and experimental alterations of the developmental process. It is thus shown that gene interactions can be modeled as a multi-dimensional algebra, and that only two interacting genes are sufficient for (i) complex pattern formation, (ii) chaotic differentiation behavior, and (iii) production of sharp edges from a continuous positional information field. This model is meant to elucidate the properties of the process of positional information-guided biomorphogenesis, not to serve as a simulation of any particular organism's development. Good quantitative data are not currently available on the interplay of gene products in morphogenesis. Thus, no attempt is made to link the images produced with actual pictures of any particular real organism. A brief introduction to top-down models and positional information is followed by the formal definition of the model. Then, the implications of the resulting morphologies to biological development are discussed, in terms of static shapes, parametrization studies, time series (movies made from individual frames), and behavior of the model in light of experimental perturbations. All figures (in grayscale), formulas and parameter values needed to re-create the figures and movies are included.